Crystal Chem ? Crystallography

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Crystal Chem ? Crystallography

• Chemistry behind minerals and how they are
  assembled
• Bonding properties and ideas governing how atoms
  go together
• Mineral assembly precipitation/ crystallization
  and defects from that
• Now we will start to look at how to look at, and
  work with, the repeatable structures which define
  minerals.
• This describes how the mineral is assembled on a
  larger scale

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Symmetry
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Symmetry Introduction

• Symmetry defines the order resulting from how
  atoms are arranged and oriented in a crystal
• Study the 2-D and 3-D order of minerals
• Do this by defining symmetry operators (there are
  13 total) ? actions which result in no change to
  the order of atoms in the crystal structure
• Combining different operators gives point groups
  which are geometrically unique units.
• Every crystal falls into some point group, which
  are segregated into 6 major crystal systems

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2-D Symmetry Operators

• Mirror Planes (m) reflection along a plane

A line denotes mirror planes
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2-D Symmetry Operators

• Rotation Axes (1, 2, 3, 4, or 6) rotation of
  360, 180, 120, 90, or 60º around a rotation axis
  yields no change in orientation/arrangement

2-fold
3-fold
4-fold
6-fold
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2-D Point groups

• All possible combinations of the 5 symmetry
  operators m, 2, 3, 4, 6, then combinations of
  the rotational operators and a mirror yield 2mm,
  3m, 4mm, 6mm
• Mathematical maximum of 10 combinations

4mm
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3-D Symmetry Operators

• Mirror Planes (m) reflection along any plane in
  3-D space

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3-D Symmetry Operators

• Rotation Axes (1, 2, 3, 4, or 6 a.k.a. A1, A2,
  A3, A4, A6) rotation of 360, 180, 120, 90, or
  60º around a rotation axis through any angle
  yields no change in orientation/arrangement

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3-D Symmetry Operators

• Inversion (i) symmetry with respect to a point,
  called an inversion center

1
1
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3-D Symmetry Operators

• Rotoinversion (1, 2, 3, 4, 6 a.k.a. A1, A2, A3,
  A4, A6) combination of rotation and inversion.
  Called bar-1, bar-2, etc.
• 1,2,6 equivalent to other functions

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• 1 Rotate 360/4

13
3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• 1 Rotate 360/4
• 2 Invert

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• 1 Rotate 360/4
• 2 Invert

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• 3 Rotate 360/4

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• 3 Rotate 360/4
• 4 Invert

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• 3 Rotate 360/4
• 4 Invert

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• 5 Rotate 360/4

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• 5 Rotate 360/4
• 6 Invert

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• This is also a unique operation

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• d. 4-fold rotoinversion ( 4 )
• A more fundamental representative of the pattern

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3-D Symmetry

• New Symmetry Elements
• 4. Rotoinversion
• c. 3-fold rotoinversion ( 3 )
• This is unique

3
5
1
4
2
6
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3-D Symmetry Operators

• Mirror planes - rotation axes (x/m) The
  combination of mirror planes and rotation axes
  that result in unique transformations is
  represented as 2/m, 4/m, and 6/m

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3-D Symmetry

• 3-D symmetry element combinations
• a. Rotation axis parallel to a mirror
• Same as 2-D
• 2 m 2mm
• 3 m 3m, also 4mm, 6mm
• b. Rotation axis ? mirror
• 2 ? m 2/m
• 3 ? m 3/m, also 4/m, 6/m
• c. Most other rotations m are impossible

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Point Groups

• Combinations of operators are often identical to
  other operators or combinations there are 13
  standard, unique operators
• I, m, 1, 2, 3, 4, 6, 3, 4, 6, 2/m, 4/m, 6/m
• These combine to form 32 unique combinations,
  called point groups
• Point groups are subdivided into 6 crystal
  systems

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3-D Symmetry

• The 32 3-D Point Groups
• Regrouped by Crystal System
• (more later when we consider
  translations)

Table 5.3 of Klein (2002) Manual of Mineral
Science, John Wiley and Sons
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Hexagonal class
Rhombohedral form
Hexagonal form
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Crystal Morphology (habit)

• Nicholas Steno (1669) Law of Constancy of
  Interfacial Angles

Quartz
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Crystal Morphology

• Diff planes have diff atomic environments

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Crystal Morphology

• Growth of crystal is affected by the conditions
  and matrix from which they grow. That one face
  grows quicker than another is generally
  determined by differences in atomic density along
  a crystal face
• Note that the internal order of the atoms can be
  the same but the crystal habit can be different!

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Crystal Morphology

• How do we keep track of the faces of a crystal?
• Face sizes may vary, but angles can't
• Thus it's the orientation angles that are the
  best source of our indexing

Miller Index is the accepted indexing method It
uses the relative intercepts of the face in
question with the crystal axes
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Miller Indices